# Which statistical method should be used in the study of stability of an enzyme?

I did a study monitoring the stability of an enzyme exposed to different conditions over time. Each day for 30 days, an aliquot was taken from each storage sample and analyzed in duplicate. I wanted to use statistical analysis to show mean changes during the 30 day experiment between the different conditions for analysis.

Here are the statistical methods I was thinking to use:

• For condition, mean value and standard deviation (SD) values at each day will be calculated and plotted versus day
• Day 1 values of mean value will be compared across the 8 conditions using a one factor analysis of variance (ANOVA) to determine if the eight methods had comparable starting value.
• If the starting value was significantly different, then the change from Day 1 will be calculated and used for comparison of the conditions. A plot of means and SDs of change in value from Day 1 by method and day will be made
• Comparison of values from the 8 conditions across the 30 days will be made using a two factor ANOVA, with conditions and days as the two factors.
• If a significant interaction of condition by day was observed, then a one-factor ANOVA will be used for each day to compare the conditions. A p value ≤ 0.05 will be used as statistically significant
• To determine deterioration in the sample based on the condition, a one-factor ANOVA will be used to make comparisons among days for each condition. This will be followed by a one-tailed Dunnett’s test to compare Days 2-30 to Day 1. There will be up to 29 comparisons for each method (30 days-1). Some of the conditions may have less than 30 days due to the condition the sample is placed in making it unable to be tested. A reasonable experiment-wise error rate of p value < 0.01 will be used.

I think you should consider multi-level models to account for the dependence of the data; you could use a random-intercept and possibly random-slope as well.

May I encourage you to take a step back and think about how your analysis will eventually be used? If I were in charge of some process that used enzymes (DNA sequencing, fruit peeling, whatever), I don't think I'd be terribly surprised to learn that enzymes degrade over time.

It's probably more useful to know how fast the enzymes go bad and how much this rate is affected by storage conditions. This information might help determine when to order/produce a new batch of enzymes or how much enzyme solution is necessary to achieve a certain effect if the solution depending on how old it is.Similarly, even if there's a significant difference, it might not be worth the hassle if the solution kept under carefully controlled conditions is 2-3 percent better than the solution kept on the counter.

The ANOVAs will tell you if there's a significant difference between conditions/days, but not the size of the difference. Instead, I'd recommend doing some regressions. You could fit each condition's data to an exponential and then compare the decay parameters (vs 0 tells you if the enzyme is degrading; vs each other to look for effects of condition). Other families of functions might be more appropriate, but this seems like a reasonable starting point to me.

A few asides:

• If I understand you correctly, you've got 2 data points per condition/day. It doesn't make a whole lot of sense to me to plot mean + standard deviation of two numbers. Maybe just plot them both?

• I'd also be tempted to normalize all the data by its starting value even if the initial ANOVA isn't significant since I think it would make the data easier to interpret. On the other hand, this might mask some kinds of changes (e.g., a contaminated sample might have a low starting value AND get worse faster than the others).